Playing Chris Kyle in “American Sniper,” the blockbuster film based on the legendary Navy SEAL’s life, changed actor Bradley Cooper’s life so profoundly that he said he probably won’t fully realize its full impact until later in his life.

While attending the annual Oscar Nominees Luncheon in Beverly Hills, Cooper was asked by reporters if he was surprised by the contentious debate that the movie has stirred about war and the military across America. Cooper avoided getting political on the subject, instead taking about how the role en...